Contemporary Mathematics
Volume 518, 2010
Modular Curves and Coding Theory: A Survey
Wen-Ching W. Li
Abstract. In this survey article we explain the role modular curves played
in the theory of error-correcting codes. The emphasis is on Elkies’ modularity
conjecture which predicts that all asymptotically optimal recursively defined
towers of curves over finite fields with square cardinality arise from reductions
of modular curves. The known examples support this conjecture. We discuss
in detail key properties of the modular towers concerning this conjecture.
1. Introduction
In the past two decades, tremendous progress has been made in the theory
of modular forms, including the proof of the Taniyama-Shimura-Weil modularity
conjecture, which was a key ingredient in the proof of the Fermat’s Last Theorem,
and the Sato-Tate conjecture. At the same time, novel applications of the theory of
modular forms have been found. One such example is the explicit construction of
Ramanujan graphs by Margulis [Mag88] and, independently, Lubotzky-PhillipsSarnak [LPS88]. They are optimal expanders which have a wide variety of applications in computer science. These graphs are sparse and highly connected at the
same time.
In coding theory, good codes should have high information rate and also high
error-correction rate. Like expanders, these are two mutually repelling features.
Many good codes are known to be algebraic geometry codes. When the base field Fq
has square cardinality, asymptotically optimal recursively defined towers of curves
(or function fields) were explicitly constructed by Garcia- Stichtenoth and company
[GS95, GS96a, GS96b, GSR03, GST97]. By properly choosing divisors, these
curves give rise to explicitly constructed good codes with arbitrarily long length.
Elkies observed that the optimal towers constructed by Garcia and Stichtenoth all
arose from reductions of elliptic modular curves, Shimura modular curves, or Drinfeld modular curves (cf. [El97, El00]). Based on these examples, he predicted that
this should hold in general, known as the Elkies’ modularity conjecture. (See §6
for details.) The work of Li-Maharaj-Stichtenoth [LMS02] provided numerical evidence to Elkies’ conjecture. This conjecture, if established, shows another instance
1991 Mathematics Subject Classification. Primary 11G20, 94B27, 11T71.
Key words and phrases. algebraic geometry codes, modular curves, towers, Elkies’ modularity
conjecture.
Research supported in part by the NSF grant DMS-0801096 and the National Center for
Theoretical Sciences, Hsinchu, Taiwan.
1
c
2010
American Mathematical Society
301
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WEN-CHING W. LI
that optimal objects possessing simultaneously repelling features arise from modular forms/modular curves. It confirms once more the general belief that number
theory provides a rich source for constructing optimal objects.
The purpose of this survey is to explain the connections between good codes
and modular curves. This paper is arranged as follows. Basic concepts in errorcorrecting codes are reviewed in §2. In §3 we introduce the Goppa codes and
algebraic geometry (AG) bound, and also discuss recent improvements on the AG
bound. The main ingredient of the AG bound is the quantity A(q) introduced by
Ihara [Iha81]. The value of A(q) is known only for square q. In §4 we review various upper and lower bounds for A(q). Asymptotically optimal recursively defined
towers are the focus of §5. Explicit towers over Fq with q a square or a cube are
recalled. In addition to introducing Elkies’ conjecture, the purpose of §6 is to explain why reductions of elliptic modular curves give rise to asymptotically optimal
and recursively defined towers. In the final section, §7, we draw connections with
solutions of Picard-Fuchs differential equations attached to modular curves, following the work of Maier [Mar07]. In particular, we show how a recursive defining
equation and the splitting set of (the reduction of) an elliptic modular tower can
be read off from the functional equation obtained from the normalized analytic solutions of the normalized weak-lifts of the Picard-Fuchs differential equation of the
bottom curve along two different projections from the second curve. This viewpoint
provides a more transparent and better understanding of the modular towers.
2. Error-correcting codes
Let q be a prime power. A linear code C of length n in q symbols is a subspace
of Fnq . The information rate of C is
dim C
.
n
If C is merely a subset of Fnq , we call it a nonlinear code of length n. Its information
rate is defined in the same way with dim C replaced by logq |C|.
The distance between two elements in Fnq is the number of components in which
they differ. The minimum distance d of a code C is the shortest distance between
two distinct codewords. Thus the balls in Fnq centered at the codewords of C with
radius (d − 1)/2 are mutually disjoint. A received word falling in the region
covered by these balls is closest to a unique codeword c; when the error probability
is small, we assume this received word was sent from c, and hence we decode it as
c and thereby correct the errors occurred during the transmission. Because of this,
we say that C corrects at least (d − 1)/2 errors. The error correction rate of C is
defined to be
d
δ(C) = .
n
By definition, both r and δ lie in the interval [0, 1]. A code is considered good
if it has high information rate r and at the same time large error-correction rate
δ. Notice that r and δ are mutually conflicting measurements, since this means
choosing many vectors far apart from each other in a fixed vector space. To further
clarify their relationship, given δ ∈ [0, 1], let
r(C) =
αq (δ) = sup {r(C) | δ(C) ≥ δ}.
C
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MODULAR CURVES AND CODING THEORY: A SURVEY
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A good code with error-correction rate δ should have its information rate r as close
to αq (δ) as possible.
It follows from definition that αq is a nonincreasing function in δ. Manin
[Man81] showed that αq is continuous. It is easy to see that αq (0) = 1 and
αq (1 − 1q ) = 0 (and hence αq (δ) = 0 for δ ∈ [1 − 1q , 1]). Unfortunately, no other
value of αq (δ) is known. It satisfies the linear Plotkin upper bound
q
αq (δ) ≤ 1 −
δ.
q−1
The celebrated Gilbert-Varshamov lower bound says
(2.1)
αq (δ) ≥ 1 − Hq (δ),
where Hq (x) is the entropy function given by
Hq (x) = x logq (q − 1) − x logq x − (1 − x) logq (1 − x).
This remained the best known lower bound for a long time until Goppa introduced
algebraic geometry codes [Gpa81] which gave rise to a better lower bound, called
algebraic geometry (AG) bound, for square q ≥ 49 and δ in a certain range. The
AG bound was improved further in the 21st century. In the next section we briefly
recall this development.
3. Goppa codes and algebraic geometry bounds
3.1. Goppa codes and AG bounds. Let V be an irreducible smooth projective curve defined over Fq with genus g. Let P1 , ..., Pn be Fq -rational points on V .
Choose a divisor G disjoint from the n rational points. One has the Riemann-Roch
space
L(G) = {f : div(f ) + G ≥ 0} ∪ {0}.
Goppa introduced a linear algebraic geometry code C of length n over Fq by evaluating the Fq -rational functions f in L(G) at the Fq -rational points P1 , ..., Pn :
C = {(f (P1 ), ..., f (Pn )) : f ∈ L(G)}.
Since these rational points are away from the poles of f , each component f (Pi ) is
an element in Fq . Moreover, the assignment f → (f (P1 ), ..., f (Pn )) is a linear map
from L(G) to C. When deg G < n, this map is injective, hence the dimension k of
C is equal to dim L(G), which is ≥ deg G − g + 1 by the Riemann-Roch theorem,
and the minimum distance d of C is ≥ n − deg G since a rational function has
the same number of zeros as poles in an algebraic closure of Fq . This shows that
the information rate r(C) = nk and the error-correcting rate δ(C) = nd satisfy the
relation
1
1
r(C) + δ(C) ≥ 1 + − n .
n
g
To describe the best lower bound, denote by Nq (g) the maximum number of
rational points among all smooth irreducible genus g curves defined over Fq . Ihara
[Iha81] introduced the quantity
(3.1)
A(q) = lim sup
g→∞
Nq (g)
.
g
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Taking the limit as n tends to infinity yields the algebraic geometry (AG) bound
for the function αq :
αq (δ) + δ ≥ 1 −
(3.2)
1
.
A(q)
3.2. Improved AG bounds. In 2001 Xing [Xn01, Xn03] constructed nonlinear subcodes of AG codes by exploiting derivatives of the rational functions
at the Fq -rational points to obtain improvements of the above AG bound. His
work has stimulated a flurry of activities along the same vein. They include the
works by Elkies [El01, El03], Niederreiter-Özbudak [NO04, NO07a, NO07b],
Stichtenoth-Xing [SX05], Maharaj [Mh07a] and Yang-Qi [YQ09]. The main idea
is to properly choose the divisor G to construct an AG code first and then strategically select a subset of this AG code to trade size for increased minimum distance.
The detailed statements are often technical. Generally speaking, the improvements
are around
1
1
+ logq (1 + 3 )
αq (δ) + δ ≥ 1 −
A(q)
q
with the best lower bound given in [YQ09] of the form
αq (δ) + δ ≥ 1 −
1
2
q−1
+ logq (1 + 3 ) + logq (1 + 6 )
A(q)
q
q
for certain values of q and δ in some interval. The paper [Li07] gives more detail
on the construction of nonlinear codes improving AG bound. See also the paper by
Niederreiter [Ni] in this volume and the references therein.
4. Upper and lower bounds for A(q)
To compare the Gilbert-Varshamov bound with the AG bound, we need an
estimate of A(q). The first bound
√
A(q) ≥ q − 1 for square q
was proved in 1981 by Ihara [Iha81] by considering the reduction of Shimura curves,
and independently in 1982 by Tsfasman-Vladut-Zink [TVZ82] by considering the
reduction of elliptic modular curves. In 1983, Vladut-Drinfeld [VD83] gave the
general upper bound
√
A(q) ≤ q − 1.
√
Combined, they imply A(q) = q − 1 when q is a square. As a consequence, one
sees that for square q ≥ 49 and δ varying in an interval depending on q, the AG
bound (3.2) is better than the Gilbert-Varshamov bound (2.1).
The truth of A(q) in unknown for nonsquare q. However, some lower bounds
were obtained. By considering class field towers analogous to the Golod-Shafarevich
tower, Serre [Ser83, Ser85] gave the first lower bound estimate
A(q) > c1 log q
for a constant c1 > 0 independent of q. Then Temkine [Tem01] and Li-Maharaj
[LM02] obtained the lower bound
A(q r ) ≥ c2 r 2 log q
log q
,
log r + log q
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MODULAR CURVES AND CODING THEORY: A SURVEY
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which improves Serre’s bound when the cardinality of the finite field is a high power
of a prime. The lower bound of the form
A(q r ) ≥ cq 1/2
was established in a sequence of papers during the period 1997-99 by NiederreiterXing [NX98, NX99] and in 2002 by Li-Maharaj [LM02]. See [LM02] for a
detailed description of the actual lower bound. For cubic powers, the estimates are
much tighter. The first was given by Zink [Z85] in 1985 using degenerations of
Shimura surfaces:
2(p2 − 1)
.
A(p3 ) ≥
p+2
The extension of this result with p replaced by prime powers q was done by BezerraGarcia-Stichtenoth [BGS05] in 2005 via explicit constructions. Their result also
improves the Gilbert-Varshamov bound over Fq3 when q ≥ 7.
5. Asymptotically optimal recursively defined towers
5.1. Recursively defined towers. A recursively defined tower over Fq is a
strictly increasing tower T of function fields
F1 ⊂ F2 ⊂ F3 · · ·
satisfying the following conditions:
(1) Each Fi is a function field with the same field of constants Fq ;
(2) Each Fi+1 is a finite separable extension of Fi ;
(3) The genus g(Fi ) → ∞ as i → ∞;
(4) F1 is the rational function field F(x1 ), Fi+1 = Fi (xi+1 ) for i ≥ 1, and
there is a rational function f (X, Y ) in variables X and Y with coefficients
in Fq such that f (xi , xi+1 ) = 0 for i ≥ 1.
This tower has a geometrical interpretation. Let Vi be the smooth projective
curve over Fq with function field Fi . Geometrically, the curves {Vi } form a covering
· · · → Vi+1 → Vi → · · · → V1 = P1 .
Further, (P1 , ..., Pi ) ∈ Vi if and only if f (Pj , Pj+1 ) = 0, i.e., (Pj , Pj+1 ) ∈ V2 for
j = 1, ..., i − 1. Thus each Vi , i ≥ 2, can be imbedded in (V2 )i−1 .
5.2. Asymptotically optimal recursively defined towers. For the tower
T , define
λ(T ) = lim
i→∞
N (Vi )
number of places of degree 1 in Fi
= lim
.
i→∞
g(Vi )
genus of Fi
Here N (Vi ) denotes the number of Fq -rational points on Vi and g(Vi ) its genus.
When λ(T ) = A(q), the tower T is said to be asymptotically optimal.
Since the value of A(q) is known only for square q, it is this case that explicit
examples of recursively defined asymptotically optimal towers are constructed. We
list a few examples below. See the paper by Maharaj [Mh07b] for more details.
(1) The first optimal tower was constructed by Garcia-Stichtenoth [GS95] in
1995. The field of constants is Fq2 and the recursive function is
f (X, Y ) = (Y X)q + Y X − X q+1 .
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WEN-CHING W. LI
(2) In 1996 Garcia-Stichtenoth [GS96a] gave another optimal tower over Fq2
with
Xq
.
f (X, Y ) = Y q + Y − q−1
X
+1
It is in fact a subtower of the first tower. Both of these towers are the
so-called wild towers, meaning that the successive extensions Fi+1 over Fi
are wildly ramified. Elkies [El00] in 2000 showed that these two towers
arose from Drinfeld modular curves by reduction.
(3) Garcia-Stichtenoth [GS96b] published two optimal tame towers in 1996
over F4 and F9 , respectively. Elkies [El97] showed in 1997 that they came
from the elliptic modular curves {X0 (3n )} mod 2 and {X0 (2n )} mod 3,
respectively. Using Jacobi identity on theta functions, Solé in 2000 gave
a different proof of this fact [So00].
(4) Aided by computer search, Li-Maharaj-Stichtenoth [LMS02] in 2002 exhibited four new optimal towers over F4 , F9 , F25 , F49 , respectively, whose
recursive function f (X, Y ) is a polynomial of low degree. These are tame
towers, and not subtowers of previously known towers. Elkies [El02]
showed that these towers came from elliptic modular curves by reduction.
(5) In 2003, Garcia-Stichtenoth-Rück [GSR03] exhibited an optimal tower
over Fp2 with
X2 + 1
.
f (X, Y ) = Y 2 −
2X
They are from the elliptic modular curves {X0 (2n )} modulo any odd prime
p.
(6) In 2004 Bezerra-Garcia [BG04] found an optimal subtower of the second
tower with the recursive function
Y − 1 Xq − 1
.
−
f (X, Y ) =
Yq
X
We also exhibit some recursively defined explicit towers over Fq3 with λ(T ) =
2(q 2 −1)
q+2 .
(1) The first such tower was constructed in 2002 by van der Geer and van der
Vlugt [GV02] over F8 with
f (X, Y ) = Y 2 + Y − X + 1 + 1/X.
(2) In 2005 Bezerra-Garcia-Stichtenoth [BGS05] constructed such a tower
over Fq3 with
1−Y
Xq + X − 1
.
−
q
Y
X
The argument was simplified by Bassa-Stichtenoth [BS07].
f (X, Y ) =
6. Elkies’ modularity conjecture
6.1. The statement of Elkies’ modularity conjecture. Based on the
above examples, Elkies in 2000 proposed the following conjecture [El00]:
Elkies’ modularity conjecture. All asymptotically optimal recursively defined towers over Fq for square q arise from reductions of elliptic modular curves,
Shimura modular curves, and Drinfeld modular curves.
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MODULAR CURVES AND CODING THEORY: A SURVEY
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More precisely, among these towers, the tame ones over Fp2 are expected to
arise from reductions of elliptic modular curves, those over Fq2 for nonprime q are
expected to come from reductions of Shimura curves, and the wild towers should
arise from reductions of Drinfeld modular curves. The work [LMS02] provided
some numerical evidence to this conjecture for small squares q and low degree
recursive defining polynomials f (X, Y ).
To explain the philosophical reasons behind Elkies’ conjecture, we restrict ourselves to towers of elliptic modular curves mod p viewed as curves over Fp2 . Recall
that for a positive integer N the modular group
a b a, b, c, d, ∈ Z, N |c, ad − bc = 1
Γ0 (N ) =
c d acts on the Poincaré upper half-plane H by fractional linear transformations. The
completion of Γ0 (N )\H by adding cusps of Γ0 (N ) is a Riemann surface of genus
g0 (N ), called the modular curve X0 (N ). It is known that X0 (N ) has a model
defined over Q. Furthermore, for a prime p coprime to N , the curve X0 (N ) has
a good reduction mod p with the same genus g0 (N ). We shall first show that for
a fixed prime p, the family {X0 (N )} mod p, as N → ∞ and p not dividing N , is
asymptotically optimal over Fp2 . Then we explain why the family {X0 (n )}, n ≥ 1,
mod p coprime to is recursively defined. Put together, we see that the family
{X0 (n )}, n ≥ 1, mod p coprime to is an asymptotically optimal recursively
defined tower.
6.2. Asymptotical optimality of elliptic modular towers. We give two
ways to see why the family {X0 (N )} mod p for p coprime to N is asymptotically
optimal over Fp2 . To count the number of rational points over Fp2 , the first method
is to appeal to the zeta function
g0
(N )
Z(X0 (N )/Fp , t) =
(1 − λi (p)t + pt2 )
i=1
(1 − t)(1 − pt)
.
Here we have used the fact that the roots β and β̄ occur in pairs in the numerator of
the zeta function Z(X0 (N )/Fp , t) to combine them in a quadratic term 1 − λi (p)t +
pt2 . A simple computation shows that
g0 (N )
N (X0 (N )/Fp2 ) = 1 + p −
2
λi (p)2 + 2pg0 (N ).
i=1
So to compute N (X0 (N )/Fp2 ) we need to estimate the sum of λi (p)2 . For this, we
invoke the so-called Eichler-Shimura congruence relation, which interprets λi (p) as
the eigenvalues of the Hecke operator Tp on weight 2 cusp forms for Γ0 (N ). This
then allows us to apply the trace formula for the trace of the Hecke operator Tp2
on such cusp forms to conclude
g0 (N )
λi (p)2 = (p + 1)g0 (N ) + o(g0 (N )).
i=1
Put together, this shows that, when N goes to infinity, the main term of N (X0 (N )/Fp2 )
is (p − 1)g0 (N ) and the error term divided by the genus g0 (N ) goes to zero. Thus
the limit of N (X0 (N )/Fp2 )/g0 (N ) is p − 1, which is precisely A(p2 ).
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WEN-CHING W. LI
The second way is to compute Fp2 -rational points on X0 (N )/Fp2 using the
moduli interpretation of the curve X0 (N ). There are two kinds of Fp2 -rational
points on this curve: the first kind is the cusps, which are negligible compared to
the genus g0 (N ); the second kind corresponds to supersingular elliptic curves over
Fp2 , whose cardinality is well-known. This is the approach in [TVZ82].
6.3. Recursively defined elliptic modular towers. In this subsection we
consider the family {X0 (n )} for > 1 and n ≥ 1. Our purpose is to show that
this family can be constructed recursively, as explained by Elkies [El00]. Consequently, modulo primes p coprime to , we obtain families of asymptotically optimal
recursively defined towers over Fp2 .
A point z ∈ Y0 (n ) = X0 (n ) − {cusps} = Γ0 (n )\H represents an equivalence
class of the pairs (E, Cn ) of elliptic curves E over C together with a cyclic subgroup
Cn of order n . We may also view this pair as an isogeny from E → E/Cn . Being
a cyclic group, Cn has a unique filtration of cyclic subgroups of the following type
Cn ⊃ Cn−1 ⊃ · · · ⊃ C .
In terms of isogenies, this yields a chain
E0 = E −→ E1 = E/C −→ · · · −→ En = E/Cn .
Hence this gives rise to an imbedding
πn :
Y0 (n ) −→ (Y0 (2 ))n−1
sending E = E0 −→ E1 −→ · · · −→ En to the point (E0 → E1 → E2 , E1 → E2 →
E3 , · · · , En−2 → En−1 → En ).
As a point in H, z represents the isogeny from E = C/(Z + zZ) to E/Cn =
C/(−n Z + zZ) ∼
= C/(Z + n zZ). So in terms of points in H, πn maps z ∈ Y0 (n ) to
the point (z, z, ..., n−2 z) ∈ (Y0 (2 ))n−1 .
It remains to find the recursive relation f (X, Y ) which characterizes the pair
(z, z) for z, z ∈ Y0 (2 ). Recall the Atkin-Lehner involution wn on X0 (n ) which
sends z to −1
n z , or equivalently, it maps E0 → E1 → · · · → En to the dual isogeny
En → En−1 → · · · → E0 . From the diagram
w 2 :
w :
X0 (2 ) −→ X0 (2 )
↓ proj
↓ proj
X0 () −→
X0 ()
we get
proj ◦ w2 (z) = −1/2 z = w ◦ proj(z).
This gives rise to the recursive function f (X, Y ).
When X0 (2 ) has genus zero, parametrize the points on the curve by a Hauptmodul x. The recursive function is a relation between x(z) and x(z); the variables
added to define the corresponding function fields in the tower are x(i z) for i ≥ 1.
Elkies in [El00] computed the following recursive defining equations:
f (X, Y ) = (X 2 − 1)((Y + 3)2 /(Y − 1)2 − 1) − 1
for = 2,
f (X, Y ) = (X 3 − 1)((Y + 2)3 /(Y − 1)3 − 1) − 1
for = 3,
f (X, Y ) = (X − 1)((Y + 1) /(Y − 1) − 1) − 1
for = 4.
4
4
4
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MODULAR CURVES AND CODING THEORY: A SURVEY
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Note that a tower can be defined by different recursive functions f (X, Y ). For
these three towers, recursive functions were also derived from functional equations
satisfied by certain hypergeometric functions 2 F1 (α, β; 1; ·) obtained from AGM
(arithmetic and geometric means) iterations and identities on theta functions. This
line of work was initiated by Gauss, and continued by Ramanujan, Borwein-Borwein
[B-B91], and Solé [So00].
6.4. Splitting set and ramification locus. Given a tower T , Garcia and
Stichtenoth introduced the splitting set of T to be
S = {degree 1 place v of F1 | v splits completely in all Fi }
and the ramification locus of T to be
R = {places v of F1 | v ramifies in some Fi over F1 }.
They showed that if T is a tame tower, then there is a convenient lower bound for
λ(T ) in terms of the S and R:
λ(T ) ≥
2g(F1 ) +
2|S|
v∈R
deg v − 2
.
In particular, when the tower T is recursively defined by f (X, Y ), choose S
to be the subset of points on the projective line P1 (Fq ) such that for each P ∈ S,
all roots of f (P, Y ) = 0 are contained in S. Then S is certainly contained in the
splitting set of T .
How does one get the splitting set and ramification locus of the modular tower
{X0 (n )} mod p?
One example was given by Garcia-Stichtenoth-Rück in their 2003 paper [GSR03],
where they showed that the tower {X0 (2n )} over Fp2 for p odd has
X2 + 1
.
2X
As the places contained in its ramification set and splitting set turn out to have
degree 1, they are identified with Fp2 -rational points as follows:
√
R = {0, ∞, ±1, ± −1},
f (X, Y ) = Y 2 −
and
S = {α ∈ F̄p : Hp (α4 ) = 0} ⊂ Fp2 .
Here
(6.1)
Hp (X) =
0≤j≤(p−1)/2
2
(p − 1)/2
Xj
j
is the Deuring polynomial. As a subtower, the tower {X0 (4n )} over Fp2 has the
same splitting set and ramification locus. More similar examples can be found in
Hasegawa [Ha07].
In the next section, we show how the recursive defining function f and the
polynomial Hp giving rise to the splitting set above arise from a functional equation
satisfied by 2 F1 ( 21 , 12 ; 1; ·). We shall arrive at this conclusion by considering PicardFuchs differential equations attached to modular curves.
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WEN-CHING W. LI
7. Connections with Picard-Fuchs differential equations
7.1. Picard-Fuchs differential equations attached to modular curves.
On the modular curve X0 (1) = P SL2 (Z)\H∗ with Hauptmodul J = 1728/j, choose
the well-known Picard-Fuchs differential equation
1
1
5/144
]DJ +
.
L1 : DJ2 + [ +
J
2(J − 1)
J(J − 1)
It has three regular singular points at J = 0 (corresponding to the cusp ∞), J = 1
(corresponding to the elliptic point of order 2), and J = ∞ (corresponding to the
elliptic point of order 3). The characteristic exponents at these points are 0, 0; 0,
1/2; 1/12, 5/12, respectively. In a neighborhood of the cusp ∞, the solution space
has as a basis one analytic function and one with logarithmic singularity at ∞. The
function h1 = 2 F1 (1/12, 5/12; 1; J(z)) is the unique analytic solution with value 1
at ∞; it is a modular form for SL2 (Z) of weight 1 with multiplier.
For a subgroup Γ of SL2 (Z) of finite index, the modular curve X(Γ) admits
Picard-Fuchs differential operators. These are second order differential equations
whose singular points are regular singular. One way to construct them is to ”weaklift” L1 along the natural projection from X(Γ) to X0 (1). Impose some normalization conditions similar to what we have observed on L1 . In particular, we demand
that the regular singular points are the cusps and the elliptic points of X(Γ).
Given > 1 there are two morphisms from X0 (2 ) to X0 () given by φ = proj
and φ = w−1 ◦ proj ◦ w2 so that φ (z) = z and φ (z) = z. To proceed, we follow
the steps described in Maier [Mar07] :
• Compatibly weak-lift L1 to a Picard-Fuchs differential equation LN on
X0 (N ) along the projection from X0 (N ) to X0 (1) for N = and 2 . Let
hN denote the unique analytic solution in a neighborhood of the cusp ∞
with the constant term 1; it is a weight 1 modular form with multiplier
for Γ0 (N ) as before.
Remark 7.1. In general, if the curve X0 (N ) has n singular points and genus
g, in addition to the characteristic exponents at each singular point, there are still
”n − 3 + 3g” projective accessory parameters and g affine accessory parameters
that can vary. Consequently, once the characteristic exponents are specified, such a
differential equation LN is unique when n = 3 and g = 0, and nonunique otherwise.
• Find a suitable weak lift of L to L2 on X0 (2 ) along the map φ such
that the two unique normalized analytic solutions h2 and h2 around the
cusp ∞ agree.
• Express h2 and h2 using h to get a functional equation of h , from which
the recursive function f (X, Y ) and the splitting set S can be read off.
For = 2, 3, 4, 5, 6, 7, the modular curve X0 () has genus zero. Choose a
Hauptmodul x with div(x ) = ∞ − 0 supported at two cusps. Further, for
= 2, 3, 4, 5, the curve X0 (2 ) also has genus zero. Choose a Hauptmodul x2 such
that div(x2 ) = ∞2 − 02 . By checking ramifications, one finds x ◦ φ = R (x2 )
for a polynomial R of degree , and x ◦ φ = x2 /S (x2 ) for a polynomial S of
degree − 1. Maier [Mar07] showed that for = 2, 3, 4, 5, the relation h2 = h2
can be expressed using h as follows:
h (R (x2 )) = [S (x2 )/S (0)]−ψ()/12 h (
x2
),
S (x2 )
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MODULAR CURVES AND CODING THEORY: A SURVEY
311
11
where ψ() is the index of Γ0 () mod center in P SL2 (Z).
For = 2, 3, 4, the total number of cusps and elliptic points for Γ0 () is three.
So its Picard-Fuchs differential equation L has three regular singular points, and h
can be expressed in terms of the hypergeometric function 2 F1 . After suitable change
of variables, one obtains the following functional equations for 2 F1 corresponding
to = 2, 3, 4, respectively:
1−x 2
) ) = (1 + 3x)1/2 2 F1 (1/4, 3/4; 1; x2 ),
(a) 2 F1 (1/4, 3/4; 1; 1 − ( 1+3x
1−x 3
(b) 2 F1 (1/3, 2/3; 1; 1 − ( 1+2x ) ) = (1 + 2x)1/2 2 F1 (1/3, 2/3; 1; x3 ),
4
2
4
(c) 2 F1 (1/2, 1/2; 1; 1 − ( 1−x
1+x ) ) = (1 + x) 2 F1 (1/2, 1/2; 1; x ).
They agree with the functional equations obtained from AGM. This gives a modular
interpretation of the functional equations.
7.2. The recursive equation and splitting set. Now we explain how to
read off the recursive equation f and the splitting set S for the elliptic modular
tower {X0 (n )} modulo a prime p coprime to . We study the case = 4 and relate
it to the known result in [GSR03] discussed at the end of §6.
Consider hypergeometric functions modulo p. In the literature this is often
treated by taking truncations to arrive at polynomials of degree < p. But then
the functional equations no longer hold. Our observation is that a more natural
approach is to raise the functions mod p to the power 1 − p. In so doing, we not
only obtain polynomials of degree < p, but also retain the functional equations. To
illustrate this point, we prove
Theorem 7.2. For odd primes p we have
2 F1 (1/2, 1/2; 1; x)
1−p
≡ Hp (x) (mod p),
where Hp is the Deuring polynomial (6.1).
Proof. The hypergeometric function
1
32
c n xn = 1 + 2 x + 2 2 x2 + · · · ,
2 F1 (1/2, 1/2; 1; x) =
2
4 2
n≥0
where
1
1 1
cn = ( ( + 1) · · · ( + n − 1)/n!)2 for n ≥ 1,
2 2
2
is a formal power series with coefficients in the ring of p-adic integers Zp which
converges to a p-adically analytic function on pZp . One checks easily that the
initial p terms in the series satisfy
cn xn ≡ Hp (x)
(mod pZp ).
0≤n≤p−1
Further, it is not hard to compute cn modulo pZp for n ≥ p to obtain
2 F1 (1/2, 1/2; 1; x)
2
≡ Hp (x)Hp (xp )Hp (xp ) · · ·
≡ Hp (x)Hp (x) Hp (x)
p
p2
···
(mod pZp )
(mod pZp ).
2
Note that the series Hp (x)Hp (x)p Hp (x)p · · · converges to an invertible function
h(x) ∈ Zp [[x]] with h(0) = 1 so that
2 F1 (1/2, 1/2; 1; x)
= h(x)(1 + pg(x))
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312
12
WEN-CHING W. LI
for some function g(x) ∈ Zp [[x]]. Thus
2 F1 (1/2, 1/2; 1; x)
p
= h(x)p (1 + pg(x))p
and hence
2 F1 (1/2, 1/2; 1; x)
1−p
=
h(x)
(1 + pg(x))1−p = Hp (x)(1 + pk(x))
h(x)p
for some k(x) ∈ Zp [[x]]. Modulo p yields the desired congruence.
Combined with (c) we obtain the functional equation for Hp :
Hp (1 − (
1−x 4
) ) ≡ (1 + x)2(1−p) Hp (x4 )
1+x
(mod p).
1−α 4
4
4
So if α4 is a root of Hp and α = −1, then so is 1 − ( 1−α
1+α ) . Write β = 1 − ( 1+α )
4
n
so that Hp (β ) = 0. Repeating this procedure, we obtain a tower ({X0 (4 )} mod
p) recursively defined by
1−X 4
) ,
1+X
which records the relation between α and β. This tower has splitting set S = {α ∈
F̄p : Hp (α4 ) = 0}. The smallest field containing S can be shown to be Fp2 .
For = 5, 6, 7 the Picard-Fuchs differential equation has 4 regular singular
points, h can be expressed using the local Heun function Hl. Maier [Mar07] also
obtained explicit functional equations in this case. Similar functional equations can
be derived for {X0 (mn )}n≥1 , where m and are coprime.
f (X, Y ) = Y 4 − 1 + (
The paper by Beelen and Bouw [BB05] explained the tower [GSR03] by
Garcia-Stichtenoth-Rück along the same line, by considering the Picard-Fuchs differential equation in characteristic p directly.
References
P. Beelen and I. Bouw, Asymptotically good towers and differential equations, Compositio Math. 141 (2005), 1405-1424.
[BS07]
A. Bassa and H. Stichtenoth, A simplified proof for the limit of a tower over a cubic
finite field, J. Number Theory 123 (2007), no. 1, 154–169.
[BG04]
J. Bezerra and A. Garcia, A tower with non-Galois steps which attains the DrinfeldVladut bound, J. Number Theory 106 (2004), no. 1, 142–154.
[BGS05] J. Bezerra, A. Garcia and H. Stichtenoth, An explicit tower of function fields over cubic
finite fields and Zink’s lower bound, J. Reine Angew. Math. 589 (2005), 159–199.
[B-B91] J. M. Borwein and P. B. Borwein, A cubic counterpart of Jacobi’s identity and the
AGM, Trans. AMS 323 (1991), No. 2, 691-701.
[El97]
N. D. Elkies, Explicit modular towers, Proceedings of the Thirty-Fifth Annual Allerton
Conference on Communication, Control and Computing, T. Basar and A. Vardy, eds.
(1997), 23-32.
[El00]
N. D. Elkies, Explicit towers of Drinfeld modular curves. Proceedings of the 3rd European Congress of Mathematics, Barcelona, 7/2000.
[El01]
N. D. Elkies, Excellent nonlinear codes from modular curves. In STOC’01: Proceedings
of the 33rd Annual ACM Symposium on Theory of Computing, Hersonissos, Crete,
Greece, pp. 200-208.
[El02]
N. D. Elkies, Appendix to New optimal tame towers of function fields over small finite
fields by W.-C. W. Li, H. Maharaj, and H. Stichtenoth, Lecture Notes in Computer
Science 2369, C.Fieker and D.R.Kohel, eds. (2002), Springer-Verlag, Berlin, pp. 384-389.
[El03]
N. D. Elkies, Still better nonlinear codes from modular curves (2003)
http://arXiv:math.NT/0308046
[BB05]
This is a free offprint provided to the author by the publisher. Copyright restrictions may apply.
MODULAR CURVES AND CODING THEORY: A SURVEY
[GS95]
[GS96a]
[GS96b]
[GSR03]
[GST97]
[GV02]
[Gpa81]
[Ha07]
[Li07]
[LMS02]
[LM02]
[LPS88]
[Iha81]
[Mh07a]
[Mh07b]
[Mar07]
[Man81]
[Mag88]
[Ni]
[NO04]
[NO07a]
[NO07b]
[NX98]
[NX99]
[Ser83]
313
13
A. Garcia and H. Stichtenoth, A tower of Artin-Schreier extensions of function fields
attaining the Drinfeld-Vladut bound, Invent. Math. 121 (1995), 211-222.
A. Garcia and H. Stichtenoth, On the asymptotic behaviour of some towers of function
fields over finite fields, J. Number Theory 61, (1996), 248-273.
A. Garcia and H. Stichtenoth, Asymptotically good towers of function fields over finite
fields, C.R. Acad. Sci. Paris Sér. I Math. 322 (1996), 1067–1070.
A. Garcia, H. Stichtenoth, and H.-G. Rück, On tame towers over finite fields, J. Reine
Angew. Math. 557 (2003), 53-80.
A. Garcia, H. Stichtenoth and M. Thomas, On towers and composita of towers of
function fields over finite fields. Finite Fields Appl. 3 (1997), no. 3, 257-274.
G. van der Geer and M. van der Vlugt, An asymptotically good tower of curves over
the field with eight elements, Bull. London Math. Soc. 34 (2002), 291-300.
V. D. Goppa, Codes on algebraic curves, Soviet Math. Dokl. 24 (1981) vol. 1, 170-172.
T. Hasegawa, On asymptotically optimal towers over quadratic fields related to Gauss
hypergeometric functions, preprint, 2007.
W.-C. W. Li, Improved algebraic geometry bounds. Recent Trends in Coding Theory and
Its Applications, AMS/IP Stud. Adv. Math., vol. 41, Amer. Math. Soc., Providence,
RI, 2007, pp. 73-81.
W.-C. W. Li, H. Maharaj, and H. Stichtenoth, New optimal tame towers of function
fields over small finite fields, Lecture Notes in Computer Science 2369, C.Fieker and
D.R.Kohel, eds. (2002), Springer-Verlag, Berlin, pp. 372-389.
W.-C. W. Li and H. Maharaj, Coverings of curves with asymptotically many rational
points, Journal of Number theory 96 (2002), 232-256.
A. Lubotzky, R. Phillips and P. Sarnak, Ramanujan graphs, Combinatorica 8 (1988)
261-277.
Y. Ihara, Some remarks on the number of rational points of algebraic curves over finite
fields, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981), 721–724.
H. Maharaj, A note on further improvements of the TVZ-bound bound, IEEE Inform.
Theory 53 (2007), no. 3, 1210–1214.
H. Maharaj, Explicit towers and codes. Recent trends in coding theory and its applications, AMS/IP Stud. Adv. Math., vol. 41, Amer. Math. Soc., Providence, RI, 2007, pp.
35–71.
R. Maier, Algebraic hypergeometric transformations of modular origin, Trans. Amer.
Math. Soc. 359 (2007), no. 8, 3859–3885.
Y. I. Manin, What is the maximal number of points on a curve over F2 ? J. Fac. Sci.
Univ. Tokyo 28 (1981), 715-720.
G. Margulis, Explicit group theoretic constructions of combinatorial schemes and their
application to the design of expanders and concentrators, J. Prob. of Info. Trans. (1988)
39-46.
H. Niederreiter, The asymptotic theory of algebraic-geometry codes, Proceedings of the
9th conference on Finite Fields and Their Applications, July 13-17, 2009, Dublin, Contemporary Math., this volume.
H. Niederreiter and F. Özbudak, Constructive asymptotic codes with an improvement
on Tsfasman-Vladut-Zink and Xing bounds, Coding, Cryptography and Combinatorics,
K. Feng, H. Niederreiter and C. Xing eds, Progress in computer Science and Applied
Logic, vol. 23, 2004, pp. 259-275.
H. Niederreiter and F. Özbudak, Further improvements on asymptotic bounds for codes
using distinguished divisors, Finite Fields Appl. 13 (2007), no. 3, 423–443.
H. Niederreiter and F. Özbudak, Improved asymptotic bounds for codes using distinguished divisors of global function fields, SIAM J. Discrete Math. 21 (2007), no. 4,
865–899.
H. Niederreiter and C.P. Xing, Towers of global function fields with asymptotically many
rational places and an improvement on the Gilbert-Varshamov bound, Math. Nachr. 195
(1998), 171–186.
H. Niederreiter and C.P. Xing, Global function fields with many rational points and
their applications, Contemporary Math. 245 (1999), 3-14.
J.-P. Serre, Sur le nombre des points rationnels d’une courbe algébrique sur un corps
fini, C.R. Acad. Sci. Paris Sér. I Math. 296 (1983), 397–402.
This is a free offprint provided to the author by the publisher. Copyright restrictions may apply.
314
14
WEN-CHING W. LI
J.-P. Serre, Rational Points on Curves over Finite Fields, Lecture Notes, Harvard
University, 1985.
[So00]
P. Solé, Towers of function fields and iterated means, IEEE Trans. Inform. Theory 46
(2000), 1532-1535.
[SX05]
H. Stichtenoth and C. Xing, Excellent codes from algebraic function fields, IEEE Trans.
on Information Theory 51 (2005), 4044-4046.
[Tem01] A. Temkine, Hilbert class field towers of function fields over finite fields and lower
bounds for A(q), J. Number Theory 87 (2001), 189–210.
[TVZ82] M.A. Tsfasman, S.G. Vladut, and T. Zink, Modular curves, Shimura curves, and Goppa
codes, better than Varshamov-Gilbert bound, Math. Nachr. 109 (1982), 21–28.
[VD83] S.G. Vladut and V.G. Drinfeld, Number of points of an algebraic curve, Funct. Anal.
Appl. 17 (1983), 53–54.
[Z85]
T. Zink, Degeneration of Shimura surfaces and a problem in coding theory, in Fundamentals of Computation Theory, L. BUDACH (ed.), Lecture Notes in Computer
Science, vol. 199, Springer, Berlin, pp. 503–511, 1985.
[Xn01]
C. P. Xing, Algebraic-Geometry codes with asymptotic parameters better than the
Gilbert-Varshamov and the Tsfasman-Vladut-Zink bounds, IEEE Trans. Inform. Theory
47 (2001), 347-352.
[Xn03]
C. P. Xing, Nonlinear codes from algebraic curves improving the Tsfasman-Vladut-Zink
bound. IEEE Trans. Inform. Theory 49 (2003), 432-437.
[YQ09] S. Yang and L. Qi, On improved asymptotic bounds for codes from global function fields,
Des. Codes Cryptogr. 53 (2009), 33-43.
[Ser85]
Department of Mathematics, Pennsylvania State University, University Park, PA
16802 U.S.A. and National Center for Theoretical Sciences, Mathematics Division,
Third General Building, National Tsing Hua University, Hsinchu, Taiwan 300, R.O.C.
E-mail address: [email protected]
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